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ISSN : 2230-7109 (Online) | ISSN : 2230-9543 (Print) IJECT Vo l . 6, Is s u e 3, Ju l y - Se p t 2015 Kinematic Principle of Reflex : A Necessary Revisit 1B.N. Biswas, 2S. Chatterjee, 3B. Choudhury, 4S Guha Mallick 1,4Sir J.C. Bose School of Engineering, SKF Group of Institutions, Hooghly, West Bengal, India 2Kanailal Vidyamandir (Fr. Section), Chandernagore, Hooghly, West Bengal, India 3Kalna College, Kalna, Burdwan, West Bengal, India

Abstract Reflex Klystron (RK) is an oscillator. Like any other oscillators, it is expected that with the switching on the power supply, the should start building up either from an infinitesimally small value or from a finite value of small excitation. Thus RK can behave as either a soft-self-excited or a hard-self-excited oscillator- yet to be explored. Naturally it takes different finite durations to attain the steady state value in these two situations. Moreover, RK is not a continuous oscillator but is a discrete time feedback oscillator and naturally the design of the plays an important role. A new method is also adopted to calculate the spectral components of the output current waveform.

Keywords Reflex Klystron Oscillator (RKO), Starting Current of RKO

I. Introduction The 1939 was the epicenter of the golden period when principle of velocity was evolved in order to overcome the transit time limitations of vacuum tubes. As a result, low power oscillator called Reflex Oscillator, a trade mark of Sperry Gyroscope Company was developed where simple RLC resonator was also replaced by ‘rhumbatrons’ meaning rhythmic oscillation with a Q value of the order of 5-88.104. The Reflex Oscillator then found useful applications in double-detection receivers or as a modulated oscillator in low power [1]. Fig. 1: Schematic Diagram of RKO

In the development of the theoretical basis of the Klystron II. in Reflex Klystron operation, kinematics were used by the W C Hahn neglecting the To explain the phenomena of in a Reflex Klystron, let space charge effects. Latter W C Hahn hinted how space charge us refer to an idealized configuration of the system as shown in effect can be taken into account to predict the tube behavior, fig. 1. The vital features of the reflex oscillators are, (1) the initial On the other hand, latter Simon Ramo developed the theory periodic wave generated when the klystron is switched on, (2) the of velocity modulation by incorporating displacement current formation of electron bunch and (3) appropriate time of return and variations of fields charge and current densities with beam of the bunches so that they transfer energy to the resonator. The cross section, length and time by invoking Maxwell’s equations. motion of electrons in the drift may be linked with the projectile He explained the principle of velocity modulation through the motion in the gravitational field of the earth. Let us refer to existence of two slow space charge waves – one of the waves the transient oscillatory voltage across the resonator grid. The propagates with a velocity slightly greater and the other slightly electron at an instant of time ‘t1’ with velocity ‘v1’ is thrown into less than that of the beam current [2]. In fact this concept helped the retarding field at the drift space, another at ‘t0’ with velocity

Webster to explain the phenomenon of de-bunching in Klystron. ‘v0’ and another at time ‘t2’ with velocity ‘v2’. By appropriately

This creates difficulty in realizing the ideal efficiency. According adjusting the repeller voltage ‘VR’ and the accelerating voltage to him de-bunching does not limit the number of bunches a beam ‘V0’, all the three electrons can be back to the resonator grid at may contain but limits length. But the author feels that the final the same time as indicated. This instant of falling back must be at conclusions more or less agree with the outcome of Kinematic the alternating voltage at the positive, thus forcing the bunch to principle of velocity modulation [3-10]. transfer energy to the resonator. This is illustrated in the modified Applegate diagram of fig. 2. Reflex Klystron Oscillator (RKO) is a velocity modulated tube and is fairly an old topic. As such a large number of papers have been published on RKO. Notwithstanding these, some of the questions still remained unanswered. A schematic arrangement of an RKO is depicted in figure 1 along with the potential distribution (neglecting space charge distribution).

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Fig. 3: Stabilization of the Oscillation Occurs When Energy Generation Equals Energy Dissipation Per Cycle.

During the growth of oscillation the effect of the transit time is taken care of by ascertaining that the returning bunch delivers energy to the buncher. But nothing is said about the effect of transit time delay on the amplitude part of oscillations. It is shown here Fig. 2: Modified Applegate Diagram for RKO and growth of that it does severely affect the stability of oscillations. Oscillation One useful form of the theory of velocity modulation has been developed by using a number of simplifying assumption, i.e., RKO is a feedback-type oscillator, where the mechanism of Kinematics. Most of them are justified for the special design and regeneration is through internal feedback by the returning beam. simple geometry of tubes used in practice. For the simplicity and Here we explain the mechanism of oscillation by invoking an ease to appreciate the bunching principle we mainly consider the electronic conductance which is non-linear function of oscillation Kinematic principle. voltage. The amount or quantum of feedback depends on the Since it is a feedback regenerative system, the poles of the transfer beam current a hint of which was given by Ginzton when he function initially must be on the right hand side of Laplace plane introduced the concept of a “starting Beam Current” [7]. Electrons (i.e. s-plane) for initializing growth of oscillation. The further which have been speeded up travel farther and take longer than they lie from the imaginary axis, the faster will be the growth. the average time to return to the resonator and electrons with For absolute stability, the poles must lie on the imaginary axis. less than average velocity require less than the average time. As Unfortunately it has to have a jitter around the imaginary axis in a result, the bunch in a reflection oscillator is formed around order to obey the principle of energy balance, i.e. energy gained the electron with average velocity which passed the resonator per cycle equals energy loss per cycle. grids when the field was zero and changing from acceleration to deceleration. This behavior is the opposite of the action in a III. Transit Time of Electron Beam double-resonator oscillator, where the bunch was formed about For simplicity we assume that all the electrons are emitted with the electrons which passed the buncher grids when the field was zero initial velocity from the cathode. They are accelerated by changing from deceleration to acceleration [11-12]. voltage V0 before they pass through the modulating voltage V

Referring to the modified Applegate diagram of RKO (fig. 2), it sin (ωt0) at the resonator. The average transit time for an electron is seen that the returning electron finds a retarding field due to the to return to the buncher-catcher is given by resonator field. The velocity of the returning electron is reduced and some of its kinetic energy is transferred to the catcher resonator (1) field. This transfer of energy not only increases the resonator rf field but also partially replenished the loss of the resonator resulting in increasing its static Q. The effective Q is considerably where, . increased. The increase of Q has two effects, viz., (i) the rate of decay of the resonator field is reduced and (ii) the bandwidth is Obviously, the dc transit time T0, when there is no rf voltage, is considerably decreased helping the existing to be quenched given by leading to spectral purity of the output waveform. This process of growth of the amplitude of oscillations continues till the cavity field is high enough to throw back the returning electron leading to some loss of energy carried by the returning electron. Thus, this process restricts the further growth of the amplitude of oscillation Thus, leading to the stable oscillation. This is illustrated in figure 3. Therefore, the time of returning of the electron to the buncher- This stabilization happens when energy gained and energy loss catcher, ‘t’ is given by per cycle is balanced. This is what is called the ‘energy balance principle’, which can be applied to find the steady state value of (2) the amplitude of oscillations.

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A. Output Current – A New Approach B. Electronic Admittance

Let a bunch of electrons during a small interval of time “dt0” It is usually convenient to express the transit time in terms of the corresponding to current I0 enter from the buncher to the drift number of oscillation cycles N. That is, space. The electrons, following a parabolic path, return to the grid of buncher within a time interval “dt”. Here “T” is the transit time.

Obviously, “t”, the time of return is given by “t = t0+T”. Hence Hence, dt dT =+1 (6) dt00 dt Again by the conservation of charge

One finds that

where ψ = ωT0. Thus we get

Since generation of oscillation is based on the synchronous supply of energy by the returning electron-bunches to the buncher, the generated waveform is inherently rich in harmonics. As such the fundamental component of the wave is calculated by Fourier analysis. But in this paper a novel method, which is simple and Fig. 5: Applegate Diagram of RKO accurate, is used to calculate the fundamental component. It can be shown that the fundamental component of current returning Referring to Applegate diagram in fig. 5, it is seen that the bunch to the buncher grid is centre is formed at the peak of the fundamental component of (3) current about the electron which leaves the resonator gap at the instant which is at a quarter of a cycle after the peak of the rf where, putting Y = X J0 (X) for convenience gap voltage. Thus taking this into consideration one writes the resonator voltage in relation to fundamental component of current 1 F (Y ) = as 0 2 1− Y 1− 1− Y 2 F(Y ) = (4) (7) Y 1− Y 2 Referring to the fundamental component of the induced currents Note that there will be a phase reversal of 1800 degree of the beam one writes for the beam admittance current and as a result the fundamental component of current induced to the resonator is

(5) Or, The results are verified through FFT calculation of the actual waveform. This is shown in fig. 4 showing remarkable (8) agreement. Again remembering that

(9) where the electronic conductance and susceptance is given by

(10)

C. Starting Current of Oscillation in RKO Fig. 4 shows the dependence of fundamental component of current on the parameter X. Taking consideration of the phase relation

Fig. 4: Theoretical and empirical results compared with FFT values the beam current (ib) injected into the resonator will be negative. as a function of modulation index X. Thus we can write that www.iject.org International Journal of Electronics & Communication Technology 11 IJECT Vo l . 6, Is s u e 3, Ju l y - Se p t 2015 ISSN : 2230-7109 (Online) | ISSN : 2230-9543 (Print)

But all the earlier papers on reflex are silent about the i b =−1.0XXX − 0.78235 + 0.353 pole movement. I0 Since RK is an oscillating system its poles, to begin with, must lie on the right half of the s-plane as far as possible for rapid growth Remembering that X= 1/2αβψ and ψ=2πN, we find that of oscillation and the poles must move towards the imaginary axis. Therefore, electronic conductance is given by The question is: does it remain on the imaginary axis? For absolute stability, the poles must lie on the imaginary axis. Unfortunately it has to have a jitter around the imaginary axis in order to obey the principle of energy balance, i.e. energy gained per cycle equals

For permitted values of sinψ, Ge becomes negative energy loss per cycle. The movements of the poles are expected to be different for soft-self and hard-self modes of oscillation. Incidentally it must be remembered that concept of poles and zeros of a system is valid for a linear system. Since an oscillator Weak oscillation corresponds to very small value of V and hence is a nonlinear circuit, we have got to equivalently linearize the X. system by invoking the technique.

Fig. 6: Beam Conductance and Cavity Load

The condition of starting oscillation is that the resonator loss along with the load will be balanced by the electronic conductance, as Fig. 7: Equivalent Circuit of a Reflex Klystron. shown in fig. 6. Thus the starting current of oscillation (Ios ) can be found as follows, An equivalent circuit of a reflex klystron is shown in fig. 7. The circuit equation is written as (noting that is the voltage across the tuned circuit)

dv 1 Gv+++= C∫ vdt I() v 0 Therefore, the value of the ‘starting current’ is given by dt L Iv()=− ( aIv − aIv35 + aIv) (11) 10 20 30

Note that higher the value of N, lower is the value of the starting current. Some idea of the value of starting current for oscillation can be had by assuming 2 dv d 35v 2 −((aI10 −+ G )v aIv20 − aIv 30 ) +=0 (12) dt dt LC Poles and zeros are concepts which are valid for linear circuits. For example, for a parallel RLC circuit, the factor σ is D. Pole Movement in RKO known to be given by . However, it can be applied to A sinusoidal oscillator can start to producing sinusoidal output a nonlinear RLC circuit, by invoking the principle of equivalent when the poles of the circuit remain in the right half plane and linearization or describing function technique in which case, the for stable oscillation there must be a mechanism that helps the damping constant can be expressed as , where V is the poles to move in right half plane [1-5]. On the other hand in an instantaneous amplitude of a sinusoidal voltage, say, V sinωt. electronic or opto- [6-10] with an embedded Here referring nonlinear electronic conductance, RLC circuit, the poles are forcibly placed on the right half plane and as far as practicable away from the imaginary axis in order to (13) help growth of oscillation as quickly as possible. And ultimately it is imagined that the poles are almost frozen on the imaginary If G is the conductance of the tank of circuit, then the total Geq=(Ge- axis. But for an ideal linear the poles are G) is written as, really frozen so that the oscillation neither grows nor decays.

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(14)

where α1, α2 and α3 are constants of the non-linear element. In the steady state when the oscillator generates periodic waveform Equation (12) turns out to be (say, with a period T) then dV2 d V −((G −+=G )V) 0 TT dt2 dt e LC = ∫∫Pdttc P dt 00 Thus it is evident or apparent that as V grows Geq (V) tends to zero in the steady state. Putting, Therefore

T dE dt = 0 ∫ dt We get, 0 Now the total energy stored in a complete cycle is written as

Considering, v(t)=V(t) ejωt, it is easily seen that, 1122 E=+ LiL Cv 22 So

dE diL dv =LiCvviiL + =+()LC dt dt dt i+= i[( aI − Gv ) + aIv35 − aIv] CL 10 20 30

dE = −+−35 ()aI10 Gv aIv20 aIv 30 ) v dt Now for a complete cycle

2p −+35 − = ∫ ()a10 I G v a20 I v a 30 I v )0 vdt 0 Assuming the output of the oscillator of the form

Fig. 8: Pole Movement in a Reflex Klystron we can write

Naturally the poles move towards the imaginary axis from the 35aI20 24 aI30 ()aI10−+ G Vosc − Vosc =0 right half of the s-or Laplace plane. Obviously in moving it will 48 intrude into the left half of the s-plane. As soon as it happens, the amplitude will start decaying because of loss of energy and again it 5a3 423a2 Vosc − Vosc −−( a10 GI /)0 = will cross to the right half plane and so it will grow again because 84 of energy gain. This process will stabilize ultimately because of the principle of energy balance when the loss per cycle equals the energy gain per cycle.

E. Amplitude of Oscillation from the ‘principle of Energy Balance’ The principle of conservation of energy states that the amount of When, energy supplied to a system during any time interval (t-t0) should be equal to the sum of the energy consumed in the system during the same time interval plus the increase of the energy stored in the system. Calling Pt the instantaneous power supplied to the system, Pc the power consumed and E0, Ethe initial and final Then energy stored, the energy equation will be given by: 2 3a2 VC02==22 5a3 It is to be remembered that for a soft-self-excited oscillator it is necessary that C1 > 0. Differentiation gives www.iject.org International Journal of Electronics & Communication Technology 13 IJECT Vo l . 6, Is s u e 3, Ju l y - Se p t 2015 ISSN : 2230-7109 (Online) | ISSN : 2230-9543 (Print)

Therefore, F. Hard-Self and Soft-Self Excitation of RKO

2 2 The governing equation of the Reflex Klystron is x = C2 + C2 + 8C1 And the output power is given by

1 2 There can be three cases of interest, as given below, P0 = VGosc 2 Case-I: when is positive: The oscillator amplitude and power output has been depicted in Here it is easily seen the oscillation will build up from an the fig. 9. infinitesimal value for and will stabilize and show a limit cycle operation.

Case-II: when C1 =0: Here again the oscillation will build up from an infinitesimal small value but will take a longer time to stabilize compared to Case I. It will also exhibit a limit cycle operation. Case-I and case-II are called ‘soft mode of self-excitation’. Case-III: When is negative Here the oscillation can build up with an initial value of such that it satisfies the condition, 2 4 - C1+β1 x - x > 0 This is known as the ‘hard mode of self excitation’. Now consider that with the system as described by the equation, the system execute oscillation of the form v = V(t) cos (ψ(t)) (17) Fig. 9: Oscillator Amplitude and Power Output Then it can be easily shown,

Referring the electrical equivalent model (fig. 7), we can write, (15) (18)

Normalizing the time by ω0 t = τ, we get,

Steady state value x0 are given by

(19) Which are obtained from Where .

Therefore, β1= 0.2076√N. Typically, α1 = 1.0, α2=0.782, α3= 0.353, Stability of the solution is obtained by considering an infinitesimal

V0= 300 and β=1. deviation from stable amplitude (x0+ξ). Substituting x0=(x0+ξ) So we can write the above equation as follows, and retaining only first order terms in ξ we get,

(16)

Where (20) Fig. 10 shows the solution in phase space, i.e., in plane. The phase space trajectory shows the growth of oscillation That is results stable oscillation, as shown and the limit cycle of this periodic, non-linear system. in fig. 11.

Fig. 10: Phase space plot of RKO, where .

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Note that tr is the normalized transit time. It is further to be noted

that it depends on N i.e. (n-1/4) when n runs through 1,2,3 etc. tr

= N/f, let f = 10 GHz (Operating frequency), the tr=(n-1/4).1/f = (n-1/4) 10-10 an I/C = I/(C.10-9), t = t.10-9, then

Typical value of that transit time, and

IV. Jump Phenomenon in RKO As explained earlier, the reflex oscillator can have both soft self- Fig. 11: Amplitude of Oscillation as a function of conductance excitation and hard-self excitation depending upon the mode

number ' N' and ratio of G / I0 (equivalent conductance of the

. timed circuit 'G' and the dc beam current ' I0 '). The zone of stability for the mode 19/4 is shown in the fig. 14. It is very interesting to observe from the diagram that there is a possibility that at any

particular value of G / I0 , there can be two values of the oscillations amplitudes, both fall in the stable zone of operation. However,

for lower value of the oscillation-amplitude the equivalent 'Qe ' is lower than that for the higher value of the amplitude of oscillations. This means that the lower amplitude of oscillation is more stable and that the higher amplitude of oscillation. To illustrate this let us

consider the particular value of G / I0 , the oscillator voltage takes

that of ‘a’. If gradually decreaseG / I0 , the point will up to ‘b’.

After that if we further try to decrease G / I0 , the point will jump to ‘c’. After this if we further decrees the path will be from c to d. This illustrates Jump Phenomena in reflex oscillator (fig. 14).

Fig. 12: Oscillation amplitude as a function of, C1, i.e., G/I0.

Fig. 14: Hysteresis and Jump Phenomena in Reflex Klystron

V. Stationary Oscillation and Its Stability Let us now consider the stability of the system when the oscillation Fig. 13: Transient Response, output amplitude (y) and dy/dt. has reached its steady state value. Incidentally, the stability of an oscillator means both the amplitude and the frequency stability G. Effect of Delay of the oscillation. The technique of finding the stability of the When the transit time delay is taken into account, the normalized stationary solution consists in seeking the solution of the amplitude amplitude equation can be expressed as and phase equations when small charges are given in the stationery www.iject.org International Journal of Electronics & Communication Technology 15 IJECT Vo l . 6, Is s u e 3, Ju l y - Se p t 2015 ISSN : 2230-7109 (Online) | ISSN : 2230-9543 (Print) state of oscillation. When the system is disturbed momentarily from VII. Discussion its stable state, let the amplitude and the phase of the oscillation It is interesting to note that when the bunches are returning they assume x0 + δx and θ0+δθ respectively. Once this happens the induce an ac voltage on the repeller. This is concluded from a paper subsequent behavior of the oscillation will be completely governed by C. J. Bakker and G. De Vries entitled “Amplification of Small by their respective amplitude and phase equations from which it Alternating Tensions by an Inductive Action of the Electrons in is easy to find the equations for ∆x and ∆θ. Thus a Valve”. The possibility of amplification is investigated

Let us assume that an increase of I0 is reflected by a change of both theoretically and experimentally when the anode is kept oscillator amplitude ∆x. To examine whether ∆x and ∆θ dies out negative, so that the electrons in a thermionic valve cannot reach with time. Let us write their incremental equations this anode, but approach it sufficiently near to induce considerable charges. The influence of the transit time is taken into account. The agreement of theory experiment is satisfactory. 2 Remembering that x0 is given by VIII. Acknowledgment The authors are grateful to the Chairman Mr B Guha Mallick of The above equation for ∆x turns out Supreme Knowledge Foundation Group of Institutions (SKFGI) for providing all possible assistance in carrying the work. The authors are also thankful to Dr Malay Kanti Dey, Scientist F, VECC, Atomic Energy Commission , Govt of India for his We have already seen that the zone of stability is defined by the assistance and advice. 2 condition x0 > β1. Therefore ∆x will die out with time. Now, noting that an increase References of I0 indicates an increase of V0. When V0 increases then transit [1] W.C.Hahn, G.F. Metcalf,“Velocity-Modulated Tubes”, time is also increase more than the mode centre value. As a result Proceedings of the I.R.E., February, 1939, pp. 106-116. the frequency of oscillation will be less than the mode centre ω0. [2] Simon Ramo,“The Electronic-Wave Theory of Velocity- Therefore, Modulation Tubes”, Proceedings of the I.R.E., December,1939, pp. 757-763. [3] Karl G. Jansky,“An Experimental Investigation of the Characteristics of Certain Types of Noise”, Proceedings of

Where (ω1 - ω0) is denoted by -Δω. This clearly indicates stability the I.R.E., December, 1939, pp. 763. of the frequency of oscillation. [4] A. E. Harrison,“Kinematics of Reflection Oscillators”, Journal of Applied Physics 15, 709, pp. 709-711, 1944. VI. Modified Reflex Klystron [5] Russell H. Varian, Stgurd F. Varian,“A High Frequency Following the idea of Bakker and Vries (cf. “Amplification Oscillator and ”, Vol. 10, May, pp. 321-327, of Small Alternating Tensions by an Inductive Action of the 1939. Electrons in a Radio Valve” C. J. Bakker and G. De Vries entitled [6] Edward Leonard Ginzton, Arthur E. Harrison,“Reflex- “Amplification of Small Alternating Tensions by an Inductive Klystron Oscillators”, Proceedings of the I.R.E. and Waves Action of the Electrons in a Radio Valve” Physica, 1934) of and Electrons, 1946, pp. 97-113. realizing amplification by inductive of the electrons from the [7] J. R. Pierce,“Reflex Oscillators”, Proceedings of the I.R.E., anode of a valve we may configure the reflector in the February, 1945, pp. 112-118. form of a resonator as shown in the adjoining figure to realize [8] Edward N. Dingley,“The Theory of Transmission Lines”, amplification. Following their analytical method it can be sown Proceedings of the I.R.E., February, pp. 118, 1945. that the output voltage as [9] J.R. Pierce,“Reflex Oscillators”, Proceedings of the I.R.E., July, 1945, pp. 483-485. [10] David L. Webster,“Cathode-Ray Bunching”, Vol. 10, July, 1939, pp. 501-508. It is interesting note that the output is proportional to frequency. [11] David L. Webster,“Velocity-Modulation Currents", Journal It means the there is a possibility of FM to AM conversion it the of Applied Physics, pp. 786-787. reflex klystron is synchronized by an FM signal. [12] David L. Webster,“The Theory of Klystron Oscillations”, Journal of Applied Physics, pp. 864-872.

Fig. 15: Modified Reflex Klystron

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